I am supposed to graph this
y= -x^2+2x
he told us to coplete the square to get us the vertex, but how do i complete the square with a - in front of the x^2??
This way:
-x^2 + 2 x = y
x^2 - 2 x = -y
x^2 - 2 x + 1 = -y+1
(x-1)^2 = -(y-1)
vertex at (1,1)
negative sign means opens down (sheds water)
To complete the square when there is a negative coefficient in front of the x^2 term, you can follow these steps:
1. Start with the equation: y = -x^2 + 2x.
2. Group the x terms together: y = -(x^2 - 2x).
3. Now, focus on the expression inside the parentheses: x^2 - 2x.
4. To complete the square, you need to add and subtract a specific constant that will allow you to write the expression as a perfect square.
5. Take half of the coefficient of the x term, which is -2/2 = -1. This will result in the equation: y = -(x^2 - 2x + (-1))^2 + 1.
6. Simplify the expression inside the parentheses: (x^2 - 2x + (-1))^2.
7. Expand the squared expression: (x^2 - 2x - 1)^2 = x^4 - 4x^3 + 6x^2 - 4x + 1.
8. Rewrite the original equation with the squared expression: y = -(x^4 - 4x^3 + 6x^2 - 4x + 1) + 1.
9. Simplify further by distributing the negative sign: y = -x^4 + 4x^3 - 6x^2 + 4x - 1 + 1.
10. Combine like terms: y = -x^4 + 4x^3 - 6x^2 + 4x.
Now, you have the equation in vertex form, y = -x^4 + 4x^3 - 6x^2 + 4x, which allows you to determine the vertex and graph the parabola.